# Different types of exponential growth equations?

• NanaToru
That was the key I was missing. Thank you!In summary, the conversation discusses a problem involving exponential growth of a bacteria culture with an initial population of 100 cells. After one hour, the population increases to 420 cells. The conversation covers finding an expression for the number of bacteria after t hours, finding the number of bacteria after 3 hours, finding the rate of growth after 3 hours, and determining when the population will reach 10,000 cells. The two common ways of modeling exponential growth, using either P(t) = Ab^t or P(t) = Aexp(kt), are also discussed.
NanaToru

## Homework Statement

A bacteria culture initially contanis 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420.
a. Find an expression for the number of bacteria after t hours.
b. find the number of bacteria after 3 hours
c. find the rate of growth after 3 hours.
d. when will population reach 10,000?​

## Homework Equations

This is done in section of the book of expontial growth and decay, so equations
dP/dt = kP
P(t) = P0 e^kt

## The Attempt at a Solution

I [/B]
I managed to solve it, but I had some questions about the exponentials.
For part a, I got
P'(t) = 100P, and if P'(1) = 420, P = 4.2

However, for c., I had to end up taking the differential of P(t) to get P'(t) = 143.5e^1.435t

Ultimately, at t = 1, these give the same answer. But I don't understand how I'm supposed to know when to get either equation... Would my answer for C have been incorrect for A?

NanaToru said:

## Homework Statement

A bacteria culture initially contanis 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420.
a. Find an expression for the number of bacteria after t hours.
b. find the number of bacteria after 3 hours
c. find the rate of growth after 3 hours.
d. when will population reach 10,000?​

## Homework Equations

This is done in section of the book of expontial growth and decay, so equations
dP/dt = kP
P(t) = P0 e^kt

## The Attempt at a Solution

I [/B]
I managed to solve it, but I had some questions about the exponentials.
For part a, I got
P'(t) = 100P, and if P'(1) = 420, P = 4.2

However, for c., I had to end up taking the differential of P(t) to get P'(t) = 143.5e^1.435t

Ultimately, at t = 1, these give the same answer. But I don't understand how I'm supposed to know when to get either equation... Would my answer for C have been incorrect for A?

The answer for C is not the same as for A.

They can't be, they have different units. Take care with your units in each part. P'(t) = 143.5e^1.435t needs units. What are they?

One is a rate (cells/hour), one is just a population (number of cells). It appears I didn't word my question right,. Thank you for your help so far.

But if A is a population equation, and C is a change in rate of population growth, why wouldn't my answer to A have been P(t) = P0e^(kt) form? My book used 100(4.2)^t for (a) and it didn't make sense to me. Is there a way that population rate of growth could've been found from that? Or were they trying to draw some comparison?

NanaToru said:
One is a rate (cells/hour), one is just a population (number of cells). It appears I didn't word my question right,. Thank you for your help so far.

But if A is a population equation, and C is a change in rate of population growth, why wouldn't my answer to A have been P(t) = P0e^(kt) form? My book used 100(4.2)^t for (a) and it didn't make sense to me. Is there a way that population rate of growth could've been found from that? Or were they trying to draw some comparison?

There are two ways of modeling exponential growth: P(t) = Ab^t, where A and b are constants and P(t) = Aexp(kt), where A and k are constants.

The way you did it is more common for bacterial growth and any time one solves the same differential equation you posted. One can use simple rules of logs and exponents to shot that the two models are equivalent and to compute the equivalent model in the other form when given one to start with.

NanaToru
Thank you for your help! I understand now.

You can use both $Aexp(kt)= Ae^{kt}$ and $Ab^t$ for the same thing because $e^x$ and $ln(x)$ are "inverse functions": $e^{ln(x)}= x$. In particular $p^t= e^{ln(p^t)}= e^{t ln(p)}= e^{kt}$ where $k= ln(p)$.

NanaToru and Dr. Courtney

## 1. What is an exponential growth equation?

An exponential growth equation is a mathematical expression that describes the growth of a quantity over time at an increasing rate. It is often represented as y = ab^x, where a is the initial value, b is the growth factor, and x is the time.

## 2. What are the different types of exponential growth equations?

There are several types of exponential growth equations, including the compound interest formula, logistic growth equation, and Malthusian growth model. Each type has its own unique characteristics and applications.

## 3. How do you solve an exponential growth equation?

To solve an exponential growth equation, you can use the properties of logarithms or take the natural log of both sides. You can also use a graphing calculator to find the intersection of the exponential function and a given value.

## 4. What are some real-world examples of exponential growth?

Exponential growth can be observed in many natural and man-made systems, such as population growth, compound interest, and the spread of infectious diseases. It is also commonly used in financial forecasting and modeling.

## 5. How is exponential growth different from linear growth?

Exponential growth is characterized by an increasing rate of change, while linear growth has a constant rate of change. In exponential growth, the quantity grows at an accelerating rate, while in linear growth, it grows at a steady rate.

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